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Theorem riotaclbg 6344
Description: Closure of restricted iota. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 24-Dec-2016.)
Assertion
Ref Expression
riotaclbg  |-  ( A  e.  V  ->  ( E! x  e.  A  ph  <->  (
iota_ x  e.  A ph )  e.  A
) )
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    V( x)

Proof of Theorem riotaclbg
StepHypRef Expression
1 riotacl 6319 . 2  |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A ph )  e.  A
)
2 undefnel2 6302 . . . 4  |-  ( A  e.  V  ->  -.  ( Undef `  A )  e.  A )
3 riotaund 6341 . . . . . 6  |-  ( -.  E! x  e.  A  ph 
->  ( iota_ x  e.  A ph )  =  ( Undef `  A ) )
43eleq1d 2349 . . . . 5  |-  ( -.  E! x  e.  A  ph 
->  ( ( iota_ x  e.  A ph )  e.  A  <->  ( Undef `  A
)  e.  A ) )
54notbid 285 . . . 4  |-  ( -.  E! x  e.  A  ph 
->  ( -.  ( iota_ x  e.  A ph )  e.  A  <->  -.  ( Undef `  A )  e.  A
) )
62, 5syl5ibrcom 213 . . 3  |-  ( A  e.  V  ->  ( -.  E! x  e.  A  ph 
->  -.  ( iota_ x  e.  A ph )  e.  A ) )
76con4d 97 . 2  |-  ( A  e.  V  ->  (
( iota_ x  e.  A ph )  e.  A  ->  E! x  e.  A  ph ) )
81, 7impbid2 195 1  |-  ( A  e.  V  ->  ( E! x  e.  A  ph  <->  (
iota_ x  e.  A ph )  e.  A
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    e. wcel 1684   E!wreu 2545   ` cfv 5255   Undefcund 6296   iota_crio 6297
This theorem is referenced by:  riotaclb  6345  riotasvd  6347  riotasvdOLD  6348  spwex  14338  supdef  25262
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-undef 6298  df-riota 6304
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